Finite Group Databasebeta

← #30599153

Subgroups — #30599153

Order 15616.

350 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461741C211yesmaximal, characteristic
2#21461367C211yesmaximal, characteristic
3#21461252C212yesmaximal
4#21460674C212yesmaximal
5#21454748C211yesmaximal, characteristic
6#16840781C2212yes—
7#16840706C2212yes—
8#16840636—21no—
9#16840636—21no—
10#16840626—21no—
11#16840623—21no—
12#16840616—22no—
13#16840615—22no—
14#16840316C2211yescharacteristic
15#16839893C411yescharacteristic, complemented
16#16839891—21no—
17#16839889C411yescharacteristic, complemented
18C8×Dic122—21no—
19C8×Dic122C411yescharacteristic, complemented
20#16839787—21no—
21#16839784C2211yescharacteristic
22#16839716—21no—
23C4×C8×D61C2211yescharacteristic
24C8×Q8×C61C411yescharacteristic, complemented
25C2×C4⋊#6923—22no—
26C4⋊#92420—22no—
27#14519834—22no—
28C4×#92420—22no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4—22no—
31C2×#11758189—21no—
32C2×C8×D61D411yescharacteristic
33C2×C8×D61D411yescharacteristic
34#14519776Q811yescharacteristic
35#14519775Q811yescharacteristic
36#14519767—21no—
37#14519757D411yescharacteristic
38#14519755—21no—
39#14519753D411yescharacteristic
40C4×Dic122—21no—
41C4×Dic122C2×C411yescharacteristic
42#14519734—21no—
43C42×D61C2311yescharacteristic
44C4⋊#92418—21no—
45C4⋊#92418C2×C411yescharacteristic
46#14519706—21no—
47#14519706—21no—
48#14519706—21no—
49#14519688—21no—
50#14519687—21no—
51C4×#92418C2×C411yescharacteristic
52C8×Dic61—21no—
53C8×Dic61D411yescharacteristic
54C8×Dic61D411yescharacteristic
55C4×Q8×C61C2×C411yescharacteristic
56C4⋊C8×C61—21no—
57C4⋊C8×C61C2×C411yescharacteristic
58C4×C8×C61—21no—
59C4×C8×C61C2×C411yescharacteristic
60C22×#6923—42no—
61C2×Q8×C61C4⋊C411yescharacteristic
62C2×C4×D61C2×D411yescharacteristic
63C2×C4×D61C2×D411yescharacteristic
64C2×C4×D61#4111yescharacteristic
65C2×Dic122—21no—
66C2×Dic122C4⋊C411yescharacteristic
67C2×#92420—42no—
68C4⋊#6923—42no—
69C4⋊#6923—42no—
70C2×C8×C61—21no—
71C2×C8×C61C22⋊C411yescharacteristic
72C2×C8×C61C22⋊C411yescharacteristic
73C4⋊C4×C61—21no—
74C4⋊C4×C61C4⋊C411yescharacteristic
75C42×C61—21no—
76C42×C61C22×C411yescharacteristic
77#11758198—21no—
78#11758198—21no—
79C4⋊Dic61—21no—
80C4⋊Dic61C4⋊C411yescharacteristic
81#11758196—21no—
82#11758196—21no—
83#11758196—21no—
84#11758196—21no—
85#11758196—21no—
86C4×Dic61—21no—
87C4×Dic61C2×D411yescharacteristic
88C4×Dic61C2×Q811yescharacteristic
89C4×Dic61#4111yescharacteristic
90C2×#92418—21no—
91C2×#92418C22⋊C411yescharacteristic
92C2×#92418C22⋊C411yescharacteristic
93#11758189—42no—
94C8×D61—22no—
95C8×D61—22no—
96C22×D61#12111yescharacteristic
97C2×#6923—82no—
98C2×#6923—82no—
99C2×#6923—82no—
100Q8×C61—41no—
101C2×C4×C61—21no—
102C2×C4×C61—21no—
103C2×C4×C61C2×C22⋊C411yescharacteristic
104C2×C4×C61C2×C4⋊C411yescharacteristic
105C2×C4×C61#11711yescharacteristic
106C2×Dic61—21no—
107C2×Dic61—21no—
108C2×Dic61#12111yescharacteristic
109C2×Dic61C22⋊Q811yescharacteristic
110C2×Dic61C22⋊Q811yescharacteristic
111C4×D61—41no—
112C4×D61—41no—
113C4×D61#13512yes—
114C4×D61#13712yes—
115Dic122—42no—
116Dic122—41no—
117#92420—82no—
118C8×C61—21no—
119C8×C61—21no—
120C8×C61—41no—
121#92418—21no—
122#92418—21no—
123#92418—41no—
124#8679—611nomaximal, Sylow
125C22×C61#38011yescharacteristic
126C2×D61—21no—
127C2×D61—21no—
128C2×D61#46412yes—
129#6923—162no—
130C4×C61—21no—
131C4×C61—21no—
132C4×C61—41no—
133C4×C61—41no—
134C4×C61#41611yescharacteristic
135C4×C61#41911yescharacteristic
136Dic61—21no—
137Dic61—21no—
138Dic61—41no—
139Dic61—41no—
140Dic61#47912yes—
141#2960—611no—
142#1835—611no—
143#1564—611no—
144#1518—612no—
145#1286—612no—
146C2×C61#180511yescharacteristic
147C2×C61#181211yescharacteristic
148C2×C61#181411yescharacteristic
149D61—22no—
150#533—611no—
151#446—611no—
152#446—611no—
153#446—1221no—
154C8×Q8—611no—
155C8×Q8—1221no—
156C8×Q8#692311yescharacteristic, complemented
157#431—612no—
158#424—1221no—
159#423—611no—
160#405—1221no—
161C2×C4×C8—611no—
162#382—612no—
163#340—1221no—
164#340—1221no—
165#335—1221no—
166#334—1221no—
167#328—1222no—
168#328—1222no—
169C61#867911yescharacteristic, complemented, Sylow
170C2×M16—1221no—
171C22×C8—611no—
172C22×C8—611no—
173C4⋊Q8—611no—
174C4⋊Q8—611no—
175#125—1221no—
176C22⋊Q8—611no—
177C22⋊Q8—611no—
178C22⋊Q8—1221no—
179C4×Q8—611no—
180C4×Q8—1221no—
181C4×Q8C2×#692311yescharacteristic
182#117—1221no—
183C2×C4⋊C4—1222no—
184C2×C42—611no—
185C4⋊C8—21no—
186C4⋊C8—611no—
187C4⋊C8—1221no—
188C4⋊C8—1221no—
189C4⋊C8—1222no—
190C4⋊C8—1221no—
191C4⋊C8—1221no—
192C4⋊C8C2×#692311yescharacteristic
193C22⋊C8—1222no—
194#97—1221no—
195#97—1221no—
196C4×C8—21no—
197C4×C8—611no—
198C4×C8—611no—
199C4×C8—611no—
200C4×C8—1222no—
201C4×C8—1221no—
202C4×C8C2×#692311yescharacteristic
203#95—1222no—
204#95—1222no—
205C2×Q8—611no—
206C2×Q8—1221no—
207C2×Q8C4⋊#692311yescharacteristic
208C22×C4—611no—
209C22×C4—611no—
210C22×C4—611no—
211C22×C4—2442no—
212M16—2442no—
213C2×C8—21no—
214C2×C8—611no—
215C2×C8—611no—
216C2×C8—1222no—
217C2×C8—1222no—
218C2×C8—1221no—
219C2×C8—2442no—
220C2×C8C22⋊#692311yescharacteristic
221C2×C8C22⋊#692311yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—611no—
224C4⋊C4—1221no—
225C4⋊C4—1221no—
226C4⋊C4—1221no—
227C4⋊C4—1221no—
228C4⋊C4—1221no—
229C4⋊C4—1221no—
230C4⋊C4—2442no—
231C4⋊C4—2442no—
232C4⋊C4C4⋊#692311yescharacteristic
233C22⋊C4—1221no—
234C22⋊C4—1221no—
235C42—21no—
236C42—611no—
237C42—611no—
238C42—611no—
239C42—1221no—
240C42C22×#692311yescharacteristic
241C23—611no—
242Q8—41no—
243Q8—2441no—
244Q8—2442no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—611no—
248C2×C4—612no—
249C2×C4—612no—
250C2×C4—611no—
251C2×C4—611no—
252C2×C4—1221no—
253C2×C4—1221no—
254C2×C4—4882no—
255C2×C4—4882no—
256C2×C4—4882no—
257C2×C4—2441no—
258C2×C4—2441no—
259C2×C4C2×C22⋊#692311yescharacteristic
260C2×C4C2×C4⋊#692311yescharacteristic
261C2×C4#1451985611yescharacteristic
262C8—21no—
263C8—21no—
264C8—41no—
265C8—1221no—
266C8—1221no—
267C8—4882no—
268C8—2441no—
269C22—612no—
270C22—1221no—
271C22—1221no—
272C22#1684070411yescharacteristic
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—612no—
278C4—1221no—
279C4—1221no—
280C4—9762no—
281C4—2441no—
282C4—2441no—
283C4#1684075211yescharacteristic
284C4#1684075511yescharacteristic
285C2—1222no—
286C2#2146169011yescharacteristic
287C2#2146170111yescharacteristic
288C2#2146170511yescharacteristic
subgroupG/H
centerC22#16840704
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC8×Q8×C61C4
socleC22×C61#380
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenterC8×Q8#6923
hypocenterC61#8679
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
15. #16839893C4semidirect
17. #16839889C4semidirect
19. C8×Dic122C4semidirect
24. C8×Q8×C61C4semidirect
156. C8×Q8#6923semidirect
169. C61#8679semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214617412
#214613672
#214612522
#214606742
#214547482
#867961